The confusion in the comments is because Hartshorne uses a definition that disagrees with the definition in EGA, as Hartshorne notes when he introduces his definition. If you use the definition in EGA, then a sufficiently positive tensor power of a relatively ample invertible sheaf on a Noetherian scheme is also relatively very ample. However, if you use the definition in Hartshorne's book, this is not true. Hartshorne includes an exercise calling attention to this disparity, and he immediately follows it by an exercise that shows, nonetheless, that a sufficiently positive tensor power of the "twist" of a relatively ample invertible sheaf on $X$ by the pullback of a sufficiently ample invertible sheaf on $Y$ does satisfy Hartshorne's definition of "relatively very ample." So Hartshorne is quite "up front" about the disparity.